MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oteqex2 Unicode version

Theorem oteqex2 4258
Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.)
Assertion
Ref Expression
oteqex2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )

Proof of Theorem oteqex2
StepHypRef Expression
1 opeqex 4257 . 2  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  _V )  <->  ( <. R ,  S >.  e.  _V  /\  T  e.  _V )
) )
2 opex 4237 . . 3  |-  <. A ,  B >.  e.  _V
32biantrur 492 . 2  |-  ( C  e.  _V  <->  ( <. A ,  B >.  e.  _V  /\  C  e.  _V )
)
4 opex 4237 . . 3  |-  <. R ,  S >.  e.  _V
54biantrur 492 . 2  |-  ( T  e.  _V  <->  ( <. R ,  S >.  e.  _V  /\  T  e.  _V )
)
61, 3, 53bitr4g 279 1  |-  ( <. <. A ,  B >. ,  C >.  =  <. <. R ,  S >. ,  T >.  ->  ( C  e.  _V  <->  T  e.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643
This theorem is referenced by:  oteqex  4259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
  Copyright terms: Public domain W3C validator